Valuation of exotic options under shortselling constraints
This article deals with the valuation of options with discontinuous payoffs. The authors use a method based on the idea of super-replication. The price of a contingent claim obtained by this method is called the upper hedging price. Let \((\Omega,{\mathcal F}(T),P)\) be a canonical probability space for a Brownian motion \(W(t)\), and let \(F(t), 0\leq t\leq T\) be the corresponding filtration. Let us denote by \(\Lambda=\{\lambda: \lambda\) is an \(\{F(t),0\leq t\leq T\}\)-adapted, nondecreasing, continuous process with \(\lambda(0)=0\}\). The following result is proved. Let \(g\) be a nonnegative lower semicontinuous function defined on \(C_{+}[0,T]\) - the space of nonnegative continuous functions on \([0,T]\). The upper hedging price for the contingent claim with payoff \(g(S)\) at the expiration date \(T\) and hedge-portfolio constraint \(\pi(t)\geq -\alpha, 0\leq t\leq T, \alpha\in [0,\infty)\) a.s. is \(v(0, S(0);\alpha)=\sup_{\lambda\in\Lambda} E\left[e^{-rT- \alpha\lambda(T)}g(Se^{-\lambda})\right]\), where \(S=\{S(t)= S(0)\exp\{\sigma W(t)+rt-\sigma^2 t/2\}, 0\leq t\leq T\}\). An application of this result for valuation of the knock-out call option is presented as well as an extension of the obtained result for the knock-out options with discrete checking of barriers. As examples the cactus option, digital put, discrete barrier option, vanilla put, lookback put, Asian put, book of two barrier options are considered.
scientific article; zbMATH DE number 1837543